Uniqueness of Non-Negative Matrix Factorization

In this paper, two new properties of stochastic vectors are introduced and a strong uniqueness theorem on non-negative matrix factorizations (NMF) is introduced. It is described how the theorem can be applied to two of the common application areas of NMF, namely music analysis and probabilistic latent semantic analysis. Additionally, the theorem can be used for selecting the model order and the sparsity parameter in sparse NMFs.